Type  Label  Description 
Statement 

Theorem  unissb 3601* 
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20Sep2003.)



Theorem  uniss2 3602* 
A subclass condition on the members of two classes that implies a
subclass relation on their unions. Proposition 8.6 of [TakeutiZaring]
p. 59. (Contributed by NM, 22Mar2004.)



Theorem  unidif 3603* 
If the difference
contains the largest
members of , then
the union of the difference is the union of . (Contributed by NM,
22Mar2004.)



Theorem  ssunieq 3604* 
Relationship implying union. (Contributed by NM, 10Nov1999.)



Theorem  unimax 3605* 
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13Aug2002.)



2.1.19 The intersection of a class


Syntax  cint 3606 
Extend class notation to include the intersection of a class (read:
'intersect ').



Definition  dfint 3607* 
Define the intersection of a class. Definition 7.35 of [TakeutiZaring]
p. 44. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 } .
Compare this with the intersection of two classes, dfin 2918.
(Contributed by NM, 18Aug1993.)



Theorem  dfint2 3608* 
Alternate definition of class intersection. (Contributed by NM,
28Jun1998.)



Theorem  inteq 3609 
Equality law for intersection. (Contributed by NM, 13Sep1999.)



Theorem  inteqi 3610 
Equality inference for class intersection. (Contributed by NM,
2Sep2003.)



Theorem  inteqd 3611 
Equality deduction for class intersection. (Contributed by NM,
2Sep2003.)



Theorem  elint 3612* 
Membership in class intersection. (Contributed by NM, 21May1994.)



Theorem  elint2 3613* 
Membership in class intersection. (Contributed by NM, 14Oct1999.)



Theorem  elintg 3614* 
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20Nov2003.)



Theorem  elinti 3615 
Membership in class intersection. (Contributed by NM, 14Oct1999.)
(Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  nfint 3616 
Boundvariable hypothesis builder for intersection. (Contributed by NM,
2Feb1997.) (Proof shortened by Andrew Salmon, 12Aug2011.)



Theorem  elintab 3617* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 30Aug1993.)



Theorem  elintrab 3618* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Oct1999.)



Theorem  elintrabg 3619* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Feb2007.)



Theorem  int0 3620 
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18Aug1993.)



Theorem  intss1 3621 
An element of a class includes the intersection of the class. Exercise
4 of [TakeutiZaring] p. 44 (with
correction), generalized to classes.
(Contributed by NM, 18Nov1995.)



Theorem  ssint 3622* 
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14Oct1999.)



Theorem  ssintab 3623* 
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31Jul2006.) (Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  ssintub 3624* 
Subclass of the least upper bound. (Contributed by NM, 8Aug2000.)



Theorem  ssmin 3625* 
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10Aug2006.)



Theorem  intmin 3626* 
Any member of a class is the smallest of those members that include it.
(Contributed by NM, 13Aug2002.) (Proof shortened by Andrew Salmon,
9Jul2011.)



Theorem  intss 3627 
Intersection of subclasses. (Contributed by NM, 14Oct1999.)



Theorem  intssunim 3628* 
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29Jul2006.)



Theorem  ssintrab 3629* 
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30Jan2015.)



Theorem  intssuni2m 3630* 
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14Aug2018.)



Theorem  intminss 3631* 
Under subset ordering, the intersection of a restricted class
abstraction is less than or equal to any of its members. (Contributed
by NM, 7Sep2013.)



Theorem  intmin2 3632* 
Any set is the smallest of all sets that include it. (Contributed by
NM, 20Sep2003.)



Theorem  intmin3 3633* 
Under subset ordering, the intersection of a class abstraction is less
than or equal to any of its members. (Contributed by NM,
3Jul2005.)



Theorem  intmin4 3634* 
Elimination of a conjunct in a class intersection. (Contributed by NM,
31Jul2006.)



Theorem  intab 3635* 
The intersection of a special case of a class abstraction. may be
free in
and , which can be
thought of a and
. (Contributed by NM, 28Jul2006.) (Proof shortened by
Mario Carneiro, 14Nov2016.)



Theorem  int0el 3636 
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24Apr2004.)



Theorem  intun 3637 
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22Sep2002.)



Theorem  intpr 3638 
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14Oct1999.)



Theorem  intprg 3639 
The intersection of a pair is the intersection of its members. Closed
form of intpr 3638. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27Apr2008.)



Theorem  intsng 3640 
Intersection of a singleton. (Contributed by Stefan O'Rear,
22Feb2015.)



Theorem  intsn 3641 
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29Sep2002.)



Theorem  uniintsnr 3642* 
The union and intersection of a singleton are equal. See also eusn 3435.
(Contributed by Jim Kingdon, 14Aug2018.)



Theorem  uniintabim 3643 
The union and the intersection of a class abstraction are equal if there
is a unique satisfying value of . (Contributed by Jim
Kingdon, 14Aug2018.)



Theorem  intunsn 3644 
Theorem joining a singleton to an intersection. (Contributed by NM,
29Sep2002.)



Theorem  rint0 3645 
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint 3646* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint2 3647* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



2.1.20 Indexed union and
intersection


Syntax  ciun 3648 
Extend class notation to include indexed union. Note: Historically
(prior to 21Oct2005), set.mm used the notation
, with
the same union symbol as cuni 3571. While that syntax was unambiguous, it
did not allow for LALR parsing of the syntax constructions in set.mm. The
new syntax uses as distinguished symbol instead of and does
allow LALR parsing. Thanks to Peter Backes for suggesting this change.



Syntax  ciin 3649 
Extend class notation to include indexed intersection. Note:
Historically (prior to 21Oct2005), set.mm used the notation
, with the
same intersection symbol as cint 3606. Although
that syntax was unambiguous, it did not allow for LALR parsing of the
syntax constructions in set.mm. The new syntax uses a distinguished
symbol
instead of and
does allow LALR parsing. Thanks to
Peter Backes for suggesting this change.



Definition  dfiun 3650* 
Define indexed union. Definition indexed union in [Stoll] p. 45. In
most applications, is independent of (although this is not
required by the definition), and depends on i.e. can be read
informally as . We call the index, the index
set, and the
indexed set. In most books, is written as
a subscript or underneath a union symbol . We use a special
union symbol to make it easier to distinguish from plain class
union. In many theorems, you will see that and are in the
same distinct variable group (meaning cannot depend on ) and
that and do not share a distinct
variable group (meaning
that can be thought of as i.e. can be substituted
with a
class expression containing ). An alternate definition tying
indexed union to ordinary union is dfiun2 3682. Theorem uniiun 3701 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27Jun1998.)



Definition  dfiin 3651* 
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union dfiun 3650. An
alternate definition tying indexed intersection to ordinary intersection
is dfiin2 3683. Theorem intiin 3702 provides a definition of ordinary
intersection in terms of indexed intersection. (Contributed by NM,
27Jun1998.)



Theorem  eliun 3652* 
Membership in indexed union. (Contributed by NM, 3Sep2003.)



Theorem  eliin 3653* 
Membership in indexed intersection. (Contributed by NM, 3Sep2003.)



Theorem  iuncom 3654* 
Commutation of indexed unions. (Contributed by NM, 18Dec2008.)



Theorem  iuncom4 3655 
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18Jan2014.)



Theorem  iunconstm 3656* 
Indexed union of a constant class, i.e. where does not depend on
. (Contributed
by Jim Kingdon, 15Aug2018.)



Theorem  iinconstm 3657* 
Indexed intersection of a constant class, i.e. where does not
depend on .
(Contributed by Jim Kingdon, 19Dec2018.)



Theorem  iuniin 3658* 
Law combining indexed union with indexed intersection. Eq. 14 in
[KuratowskiMostowski] p.
109. This theorem also appears as the last
example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29.
(Contributed by NM, 17Aug2004.) (Proof shortened by Andrew Salmon,
25Jul2011.)



Theorem  iunss1 3659* 
Subclass theorem for indexed union. (Contributed by NM, 10Dec2004.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iinss1 3660* 
Subclass theorem for indexed union. (Contributed by NM,
24Jan2012.)



Theorem  iuneq1 3661* 
Equality theorem for indexed union. (Contributed by NM,
27Jun1998.)



Theorem  iineq1 3662* 
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27Jun1998.)



Theorem  ss2iun 3663 
Subclass theorem for indexed union. (Contributed by NM, 26Nov2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2 3664 
Equality theorem for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2 3665 
Equality theorem for indexed intersection. (Contributed by NM,
22Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2i 3666 
Equality inference for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2i 3667 
Equality inference for indexed intersection. (Contributed by NM,
22Oct2003.)



Theorem  iineq2d 3668 
Equality deduction for indexed intersection. (Contributed by NM,
7Dec2011.)



Theorem  iuneq2dv 3669* 
Equality deduction for indexed union. (Contributed by NM,
3Aug2004.)



Theorem  iineq2dv 3670* 
Equality deduction for indexed intersection. (Contributed by NM,
3Aug2004.)



Theorem  iuneq1d 3671* 
Equality theorem for indexed union, deduction version. (Contributed by
Drahflow, 22Oct2015.)



Theorem  iuneq12d 3672* 
Equality deduction for indexed union, deduction version. (Contributed
by Drahflow, 22Oct2015.)



Theorem  iuneq2d 3673* 
Equality deduction for indexed union. (Contributed by Drahflow,
22Oct2015.)



Theorem  nfiunxy 3674* 
Boundvariable hypothesis builder for indexed union. (Contributed by
Mario Carneiro, 25Jan2014.)



Theorem  nfiinxy 3675* 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by Mario Carneiro, 25Jan2014.)



Theorem  nfiunya 3676* 
Boundvariable hypothesis builder for indexed union. (Contributed by
Mario Carneiro, 25Jan2014.)



Theorem  nfiinya 3677* 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by Mario Carneiro, 25Jan2014.)



Theorem  nfiu1 3678 
Boundvariable hypothesis builder for indexed union. (Contributed by
NM, 12Oct2003.)



Theorem  nfii1 3679 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by NM, 15Oct2003.)



Theorem  dfiun2g 3680* 
Alternate definition of indexed union when is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 23Mar2006.) (Proof
shortened by Andrew Salmon, 25Jul2011.)



Theorem  dfiin2g 3681* 
Alternate definition of indexed intersection when is a set.
(Contributed by Jeff Hankins, 27Aug2009.)



Theorem  dfiun2 3682* 
Alternate definition of indexed union when is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 27Jun1998.) (Revised by
David Abernethy, 19Jun2012.)



Theorem  dfiin2 3683* 
Alternate definition of indexed intersection when is a set.
Definition 15(b) of [Suppes] p. 44.
(Contributed by NM, 28Jun1998.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  dfiunv2 3684* 
Define double indexed union. (Contributed by FL, 6Nov2013.)



Theorem  cbviun 3685* 
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 26Mar2006.) (Revised by Andrew Salmon, 25Jul2011.)



Theorem  cbviin 3686* 
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26Aug2009.) (Revised by Mario Carneiro, 14Oct2016.)



Theorem  cbviunv 3687* 
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 15Sep2003.)



Theorem  cbviinv 3688* 
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26Aug2009.)



Theorem  iunss 3689* 
Subset theorem for an indexed union. (Contributed by NM, 13Sep2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun 3690* 
Subset implication for an indexed union. (Contributed by NM,
3Sep2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun2 3691 
Identity law for subset of an indexed union. (Contributed by NM,
12Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun2s 3692* 
Subset relationship for an indexed union. (Contributed by NM,
26Oct2003.)



Theorem  iunss2 3693* 
A subclass condition on the members of two indexed classes
and that implies a subclass relation on their indexed
unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59.
Compare uniss2 3602. (Contributed by NM, 9Dec2004.)



Theorem  iunab 3694* 
The indexed union of a class abstraction. (Contributed by NM,
27Dec2004.)



Theorem  iunrab 3695* 
The indexed union of a restricted class abstraction. (Contributed by
NM, 3Jan2004.) (Proof shortened by Mario Carneiro, 14Nov2016.)



Theorem  iunxdif2 3696* 
Indexed union with a class difference as its index. (Contributed by NM,
10Dec2004.)



Theorem  ssiinf 3697 
Subset theorem for an indexed intersection. (Contributed by FL,
15Oct2012.) (Proof shortened by Mario Carneiro, 14Oct2016.)



Theorem  ssiin 3698* 
Subset theorem for an indexed intersection. (Contributed by NM,
15Oct2003.)



Theorem  iinss 3699* 
Subset implication for an indexed intersection. (Contributed by NM,
15Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iinss2 3700 
An indexed intersection is included in any of its members. (Contributed
by FL, 15Oct2012.)

